every multiplication has an implied 1 is rather simple to conceptualize. Multiplication is basically "quantity x
of quantity y
." I'm pretty sure we can all agree that if you have 1 of a quantity, you have that quantity. If I, say, had 1 group of 8 apples, well, then I have those 8 apples. If I had 1 group of 3.5 bananas, I'd have 3.5 bananas.
Now, this extends to general multiplications. Say I have 1 set of 5 sets of 9 bananas each. Well, in the 5 sets of 9 bananas each, I'd have a total of 45 bananas, right? Now, if I had just 1 of this superset, then I'd still have a
set of 45 bananas. This all is the basis for the multiplicative identity.
As for your question regarding every addition having an implied zero ... well, it's a similar (but not equivalent) principle: when you add nothing to something, you end up with what you had to start with. Add 0 to 2 and you end up with 2. Add 0 to pi and you end up with pi. If I had 8 oranges and someone gave me none, I still have 8 oranges. This is otherwise known as the additive identity.
Hope this helps.
EDIT: By this same reasoning, we say that all numbers are complex numbers, just that the real numbers are complex numbers of the form a
, with a null imaginary part. But I'm getting ahead of myself, here.